91Ó°ÊÓ

Use the method of Frobenius to find solutions near \(x=0\) of each of the differential equations in Exercises, $$ 2 x^{2} y^{\prime \prime}+5 x y^{\prime}+(2 x-2) y=0 $$

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises. $$ (x-1) y^{\prime \prime}-(3 x-2) y^{\prime}+2 x y=0 $$

Find power series$$ \left(x^{3}-1\right) y^{\prime \prime}+x^{2} y^{\prime}+x y=0 $$ solutions in powers of \(x\) of each of the differential equations in Exercises.

Find power series solutions in powers of \(x\) of each of the differential equations in Exercises. $$ (x+3) y^{\prime \prime}+(x+2) y^{\prime}+y=0 $$

Find the power series solution of each of the initial-value problems in Exercises. $$ y^{\prime \prime}-x y^{\prime}-y=0, \quad y(0)=1, \quad y^{\prime}(0)=0 $$

Find the power series solution of each of the initial-value problems in Exercises. $$ y^{\prime \prime}+x y^{\prime}-2 y=0, \quad y(0)=0, \quad y^{\prime}(0)=1 $$

Find the power series solution of each of the initial-value problems in Exercises. $$ y^{\prime \prime}+x^{2} y^{\prime}+x^{2} y=0, \quad y(0)=2, \quad y^{\prime}(0)=4 $$

Find the power series solution of each of the initial-value problems in Exercises. $$ \left(x^{2}+1\right) y^{\prime \prime}+x y^{\prime}+2 x y=0, \quad y(0)=2, \quad y^{\prime}(0)=3 $$

Find the power series solution of each of the initial-value problems in Exercises. $$ \left(2 x^{2}-3\right) y^{\prime \prime}-2 x y^{\prime}+y=0, \quad y(0)=-1, \quad y^{\prime}(0)=5 $$

Find the power series solution of each of the initial-value problems in Exercises. $$ \left(x^{2}-1\right) y^{\prime \prime}+4 x y^{\prime}+2 y=0, \quad y(0)=1, \quad y^{\prime}(0)=-1 $$